LMFDB - Embedded newform 8280.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -4.77846 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.77846 q^{7} -3.30777 q^{11} +3.19051 q^{13} +2.41205 q^{17} +2.24914 q^{19} -1.00000 q^{23} +1.00000 q^{25} +7.96896 q^{29} -10.3354 q^{31} -4.77846 q^{35} -11.2767 q^{37} -1.02760 q^{41} +10.0552 q^{43} -2.74742 q^{47} +15.8337 q^{49} +5.58795 q^{53} -3.30777 q^{55} +1.96896 q^{59} -8.24914 q^{61} +3.19051 q^{65} -7.71982 q^{67} -16.0242 q^{71} -7.92332 q^{73} +15.8061 q^{77} +12.5845 q^{83} +2.41205 q^{85} +3.05863 q^{89} -15.2457 q^{91} +2.24914 q^{95} -5.43965 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 6 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 6 * q^7	$ 3 q + 3 q^{5} - 6 q^{7} - 2 q^{11} + 6 q^{17} - 2 q^{19} - 3 q^{23} + 3 q^{25} + 6 q^{29} - 6 q^{31} - 6 q^{35} - 8 q^{37} + 14 q^{41} - 4 q^{43} + 18 q^{47} + 5 q^{49} + 18 q^{53} - 2 q^{55} - 12 q^{59} - 16 q^{61} - 14 q^{67} + 4 q^{71} + 22 q^{77} + 4 q^{83} + 6 q^{85} + 10 q^{89} - 2 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 6 * q^7 - 2 * q^11 + 6 * q^17 - 2 * q^19 - 3 * q^23 + 3 * q^25 + 6 * q^29 - 6 * q^31 - 6 * q^35 - 8 * q^37 + 14 * q^41 - 4 * q^43 + 18 * q^47 + 5 * q^49 + 18 * q^53 - 2 * q^55 - 12 * q^59 - 16 * q^61 - 14 * q^67 + 4 * q^71 + 22 * q^77 + 4 * q^83 + 6 * q^85 + 10 * q^89 - 2 * q^91 - 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.77846	−1.80609	−0.903044	−	0.429549i	$-0.858673\pi$
$7$	−4.77846	−1.80609	−0.903044	+	0.429549i	$0.858673\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.30777	−0.997331	−0.498666	−	0.866794i	$-0.666176\pi$
$11$	−3.30777	−0.997331	−0.498666	+	0.866794i	$0.666176\pi$
$12$	0	0
$13$	3.19051	0.884888	0.442444	−	0.896796i	$-0.354112\pi$
$13$	3.19051	0.884888	0.442444	+	0.896796i	$0.354112\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.41205	0.585008	0.292504	−	0.956264i	$-0.405512\pi$
$17$	2.41205	0.585008	0.292504	+	0.956264i	$0.405512\pi$
$18$	0	0
$19$	2.24914	0.515988	0.257994	−	0.966146i	$-0.416938\pi$
$19$	2.24914	0.515988	0.257994	+	0.966146i	$0.416938\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.96896	1.47980	0.739900	−	0.672717i	$-0.234873\pi$
$29$	7.96896	1.47980	0.739900	+	0.672717i	$0.234873\pi$
$30$	0	0
$31$	−10.3354	−1.85629	−0.928144	−	0.372222i	$-0.878596\pi$
$31$	−10.3354	−1.85629	−0.928144	+	0.372222i	$0.878596\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.77846	−0.807707
$36$	0	0
$37$	−11.2767	−1.85388	−0.926942	−	0.375204i	$-0.877573\pi$
$37$	−11.2767	−1.85388	−0.926942	+	0.375204i	$0.877573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.02760	−0.160484	−0.0802419	−	0.996775i	$-0.525569\pi$
$41$	−1.02760	−0.160484	−0.0802419	+	0.996775i	$0.525569\pi$
$42$	0	0
$43$	10.0552	1.53340	0.766701	−	0.642004i	$-0.221897\pi$
$43$	10.0552	1.53340	0.766701	+	0.642004i	$0.221897\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.74742	−0.400753	−0.200376	−	0.979719i	$-0.564216\pi$
$47$	−2.74742	−0.400753	−0.200376	+	0.979719i	$0.564216\pi$
$48$	0	0
$49$	15.8337	2.26195
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.58795	0.767564	0.383782	−	0.923424i	$-0.374621\pi$
$53$	5.58795	0.767564	0.383782	+	0.923424i	$0.374621\pi$
$54$	0	0
$55$	−3.30777	−0.446020
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.96896	0.256337	0.128169	−	0.991752i	$-0.459090\pi$
$59$	1.96896	0.256337	0.128169	+	0.991752i	$0.459090\pi$
$60$	0	0
$61$	−8.24914	−1.05619	−0.528097	−	0.849184i	$-0.677094\pi$
$61$	−8.24914	−1.05619	−0.528097	+	0.849184i	$0.677094\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.19051	0.395734
$66$	0	0
$67$	−7.71982	−0.943127	−0.471563	−	0.881832i	$-0.656310\pi$
$67$	−7.71982	−0.943127	−0.471563	+	0.881832i	$0.656310\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.0242	−1.90172	−0.950859	−	0.309624i	$-0.899797\pi$
$71$	−16.0242	−1.90172	−0.950859	+	0.309624i	$0.899797\pi$
$72$	0	0
$73$	−7.92332	−0.927355	−0.463677	−	0.886004i	$-0.653470\pi$
$73$	−7.92332	−0.927355	−0.463677	+	0.886004i	$0.653470\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.8061	1.80127
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.5845	1.38133	0.690665	−	0.723175i	$-0.257318\pi$
$83$	12.5845	1.38133	0.690665	+	0.723175i	$0.257318\pi$
$84$	0	0
$85$	2.41205	0.261624
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.05863	0.324214	0.162107	−	0.986773i	$-0.448171\pi$
$89$	3.05863	0.324214	0.162107	+	0.986773i	$0.448171\pi$
$90$	0	0
$91$	−15.2457	−1.59818
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.24914	0.230757
$96$	0	0
$97$	−5.43965	−0.552313	−0.276156	−	0.961113i	$-0.589061\pi$
$97$	−5.43965	−0.552313	−0.276156	+	0.961113i	$0.589061\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.52932	0.848699	0.424349	−	0.905499i	$-0.360503\pi$
$101$	8.52932	0.848699	0.424349	+	0.905499i	$0.360503\pi$
$102$	0	0
$103$	−9.32238	−0.918562	−0.459281	−	0.888291i	$-0.651893\pi$
$103$	−9.32238	−0.918562	−0.459281	+	0.888291i	$0.651893\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.1449	1.27076	0.635381	−	0.772199i	$-0.280843\pi$
$107$	13.1449	1.27076	0.635381	+	0.772199i	$0.280843\pi$
$108$	0	0
$109$	10.0406	0.961714	0.480857	−	0.876799i	$-0.340326\pi$
$109$	10.0406	0.961714	0.480857	+	0.876799i	$0.340326\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.70522	0.348557	0.174279	−	0.984696i	$-0.444241\pi$
$113$	3.70522	0.348557	0.174279	+	0.984696i	$0.444241\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.5259	−1.05658
$120$	0	0
$121$	−0.0586332	−0.00533029
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.68879	0.149856	0.0749279	−	0.997189i	$-0.476127\pi$
$127$	1.68879	0.149856	0.0749279	+	0.997189i	$0.476127\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.99656	0.786033	0.393017	−	0.919531i	$-0.371431\pi$
$131$	8.99656	0.786033	0.393017	+	0.919531i	$0.371431\pi$
$132$	0	0
$133$	−10.7474	−0.931920
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.5569	1.67086	0.835430	−	0.549597i	$-0.185219\pi$
$137$	19.5569	1.67086	0.835430	+	0.549597i	$0.185219\pi$
$138$	0	0
$139$	−15.3319	−1.30044	−0.650219	−	0.759747i	$-0.725323\pi$
$139$	−15.3319	−1.30044	−0.650219	+	0.759747i	$0.725323\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.5535	−0.882526
$144$	0	0
$145$	7.96896	0.661786
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.80605	−0.803343	−0.401672	−	0.915784i	$-0.631571\pi$
$149$	−9.80605	−0.803343	−0.401672	+	0.915784i	$0.631571\pi$
$150$	0	0
$151$	14.0552	1.14380	0.571898	−	0.820325i	$-0.306207\pi$
$151$	14.0552	1.14380	0.571898	+	0.820325i	$0.306207\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.3354	−0.830157
$156$	0	0
$157$	−2.28018	−0.181978	−0.0909889	−	0.995852i	$-0.529003\pi$
$157$	−2.28018	−0.181978	−0.0909889	+	0.995852i	$0.529003\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.77846	0.376595
$162$	0	0
$163$	13.5569	1.06186	0.530930	−	0.847416i	$-0.321843\pi$
$163$	13.5569	1.06186	0.530930	+	0.847416i	$0.321843\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.8647	0.995499	0.497750	−	0.867321i	$-0.334160\pi$
$167$	12.8647	0.995499	0.497750	+	0.867321i	$0.334160\pi$
$168$	0	0
$169$	−2.82066	−0.216974
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3224	0.860825	0.430412	−	0.902632i	$-0.358368\pi$
$173$	11.3224	0.860825	0.430412	+	0.902632i	$0.358368\pi$
$174$	0	0
$175$	−4.77846	−0.361217
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.9345	1.56472	0.782359	−	0.622828i	$-0.214016\pi$
$179$	20.9345	1.56472	0.782359	+	0.622828i	$0.214016\pi$
$180$	0	0
$181$	1.93793	0.144045	0.0720226	−	0.997403i	$-0.477055\pi$
$181$	1.93793	0.144045	0.0720226	+	0.997403i	$0.477055\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.2767	−0.829082
$186$	0	0
$187$	−7.97852	−0.583447
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.7440	0.849765	0.424882	−	0.905249i	$-0.360315\pi$
$191$	11.7440	0.849765	0.424882	+	0.905249i	$0.360315\pi$
$192$	0	0
$193$	3.29317	0.237047	0.118524	−	0.992951i	$-0.462184\pi$
$193$	3.29317	0.237047	0.118524	+	0.992951i	$0.462184\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.73281	0.479693	0.239847	−	0.970811i	$-0.422903\pi$
$197$	6.73281	0.479693	0.239847	+	0.970811i	$0.422903\pi$
$198$	0	0
$199$	9.82066	0.696168	0.348084	−	0.937463i	$-0.386832\pi$
$199$	9.82066	0.696168	0.348084	+	0.937463i	$0.386832\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−38.0794	−2.67265
$204$	0	0
$205$	−1.02760	−0.0717705
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.43965	−0.514611
$210$	0	0
$211$	27.0974	1.86546	0.932731	−	0.360573i	$-0.117419\pi$
$211$	27.0974	1.86546	0.932731	+	0.360573i	$0.117419\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.0552	0.685759
$216$	0	0
$217$	49.3871	3.35262
$218$	0	0
$219$	0	0
$220$	0	0
$221$	7.69566	0.517666
$222$	0	0
$223$	20.9053	1.39992	0.699960	−	0.714182i	$-0.253201\pi$
$223$	20.9053	1.39992	0.699960	+	0.714182i	$0.253201\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	29.0518	1.91979	0.959897	−	0.280353i	$-0.0904514\pi$
$229$	29.0518	1.91979	0.959897	+	0.280353i	$0.0904514\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.5535	1.34650	0.673252	−	0.739414i	$-0.264897\pi$
$233$	20.5535	1.34650	0.673252	+	0.739414i	$0.264897\pi$
$234$	0	0
$235$	−2.74742	−0.179222
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.0862	−1.29927	−0.649635	−	0.760246i	$-0.725078\pi$
$239$	−20.0862	−1.29927	−0.649635	+	0.760246i	$0.725078\pi$
$240$	0	0
$241$	−0.574960	−0.0370364	−0.0185182	−	0.999829i	$-0.505895\pi$
$241$	−0.574960	−0.0370364	−0.0185182	+	0.999829i	$0.505895\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.8337	1.01157
$246$	0	0
$247$	7.17590	0.456592
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.32238	−0.335946	−0.167973	−	0.985792i	$-0.553722\pi$
$251$	−5.32238	−0.335946	−0.167973	+	0.985792i	$0.553722\pi$
$252$	0	0
$253$	3.30777	0.207958
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.9164	−1.11760	−0.558799	−	0.829303i	$-0.688738\pi$
$257$	−17.9164	−1.11760	−0.558799	+	0.829303i	$0.688738\pi$
$258$	0	0
$259$	53.8854	3.34828
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.9655	1.16946	0.584732	−	0.811226i	$-0.301200\pi$
$263$	18.9655	1.16946	0.584732	+	0.811226i	$0.301200\pi$
$264$	0	0
$265$	5.58795	0.343265
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.7604	0.838987	0.419494	−	0.907758i	$-0.362208\pi$
$269$	13.7604	0.838987	0.419494	+	0.907758i	$0.362208\pi$
$270$	0	0
$271$	19.1595	1.16386	0.581928	−	0.813241i	$-0.302299\pi$
$271$	19.1595	1.16386	0.581928	+	0.813241i	$0.302299\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.30777	−0.199466
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.63359	0.336072	0.168036	−	0.985781i	$-0.446258\pi$
$281$	5.63359	0.336072	0.168036	+	0.985781i	$0.446258\pi$
$282$	0	0
$283$	12.0456	0.716039	0.358020	−	0.933714i	$-0.383452\pi$
$283$	12.0456	0.716039	0.358020	+	0.933714i	$0.383452\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.91033	0.289848
$288$	0	0
$289$	−11.1820	−0.657766
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.73443	0.218168	0.109084	−	0.994033i	$-0.465208\pi$
$293$	3.73443	0.218168	0.109084	+	0.994033i	$0.465208\pi$
$294$	0	0
$295$	1.96896	0.114638
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.19051	−0.184512
$300$	0	0
$301$	−48.0483	−2.76946
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.24914	−0.472344
$306$	0	0
$307$	20.6302	1.17743	0.588713	−	0.808342i	$-0.299635\pi$
$307$	20.6302	1.17743	0.588713	+	0.808342i	$0.299635\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.1138	−1.08385	−0.541923	−	0.840428i	$-0.682304\pi$
$311$	−19.1138	−1.08385	−0.541923	+	0.840428i	$0.682304\pi$
$312$	0	0
$313$	−2.84053	−0.160556	−0.0802781	−	0.996773i	$-0.525581\pi$
$313$	−2.84053	−0.160556	−0.0802781	+	0.996773i	$0.525581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.9560	1.06467	0.532337	−	0.846533i	$-0.321314\pi$
$317$	18.9560	1.06467	0.532337	+	0.846533i	$0.321314\pi$
$318$	0	0
$319$	−26.3595	−1.47585
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.42504	0.301857
$324$	0	0
$325$	3.19051	0.176978
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.1284	0.723794
$330$	0	0
$331$	30.5078	1.67686	0.838431	−	0.545008i	$-0.183473\pi$
$331$	30.5078	1.67686	0.838431	+	0.545008i	$0.183473\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.71982	−0.421779
$336$	0	0
$337$	−0.117266	−0.00638790	−0.00319395	−	0.999995i	$-0.501017\pi$
$337$	−0.117266	−0.00638790	−0.00319395	+	0.999995i	$0.501017\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	34.1871	1.85133
$342$	0	0
$343$	−42.2112	−2.27919
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.3845	−1.14798	−0.573989	−	0.818863i	$-0.694605\pi$
$347$	−21.3845	−1.14798	−0.573989	+	0.818863i	$0.694605\pi$
$348$	0	0
$349$	3.10428	0.166168	0.0830841	−	0.996543i	$-0.473523\pi$
$349$	3.10428	0.166168	0.0830841	+	0.996543i	$0.473523\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−25.9164	−1.37939	−0.689697	−	0.724098i	$-0.742256\pi$
$353$	−25.9164	−1.37939	−0.689697	+	0.724098i	$0.742256\pi$
$354$	0	0
$355$	−16.0242	−0.850474
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.01461	0.106327	0.0531635	−	0.998586i	$-0.483070\pi$
$359$	2.01461	0.106327	0.0531635	+	0.998586i	$0.483070\pi$
$360$	0	0
$361$	−13.9414	−0.733756
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.92332	−0.414726
$366$	0	0
$367$	−12.7785	−0.667030	−0.333515	−	0.942745i	$-0.608235\pi$
$367$	−12.7785	−0.667030	−0.333515	+	0.942745i	$0.608235\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−26.7018	−1.38629
$372$	0	0
$373$	14.9345	0.773279	0.386639	−	0.922231i	$-0.373636\pi$
$373$	14.9345	0.773279	0.386639	+	0.922231i	$0.373636\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.4250	1.30946
$378$	0	0
$379$	−13.9931	−0.718779	−0.359389	−	0.933188i	$-0.617015\pi$
$379$	−13.9931	−0.718779	−0.359389	+	0.933188i	$0.617015\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.1414	0.926984	0.463492	−	0.886101i	$-0.346596\pi$
$383$	18.1414	0.926984	0.463492	+	0.886101i	$0.346596\pi$
$384$	0	0
$385$	15.8061	0.805551
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.850080	0.0431008	0.0215504	−	0.999768i	$-0.493140\pi$
$389$	0.850080	0.0431008	0.0215504	+	0.999768i	$0.493140\pi$
$390$	0	0
$391$	−2.41205	−0.121983
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.67762	0.234763	0.117381	−	0.993087i	$-0.462550\pi$
$397$	4.67762	0.234763	0.117381	+	0.993087i	$0.462550\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.4948	−1.57278	−0.786389	−	0.617732i	$-0.788052\pi$
$401$	−31.4948	−1.57278	−0.786389	+	0.617732i	$0.788052\pi$
$402$	0	0
$403$	−32.9751	−1.64261
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.3009	1.84894
$408$	0	0
$409$	−32.6803	−1.61594	−0.807968	−	0.589226i	$-0.799433\pi$
$409$	−32.6803	−1.61594	−0.807968	+	0.589226i	$0.799433\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.40861	−0.462968
$414$	0	0
$415$	12.5845	0.617749
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.926759	−0.0452751	−0.0226376	−	0.999744i	$-0.507206\pi$
$419$	−0.926759	−0.0452751	−0.0226376	+	0.999744i	$0.507206\pi$
$420$	0	0
$421$	−25.4182	−1.23881	−0.619403	−	0.785073i	$-0.712625\pi$
$421$	−25.4182	−1.23881	−0.619403	+	0.785073i	$0.712625\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.41205	0.117002
$426$	0	0
$427$	39.4182	1.90758
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.6121	−1.52270	−0.761351	−	0.648340i	$-0.775464\pi$
$431$	−31.6121	−1.52270	−0.761351	+	0.648340i	$0.775464\pi$
$432$	0	0
$433$	−27.1855	−1.30645	−0.653225	−	0.757164i	$-0.726584\pi$
$433$	−27.1855	−1.30645	−0.653225	+	0.757164i	$0.726584\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.24914	−0.107591
$438$	0	0
$439$	17.5569	0.837946	0.418973	−	0.907999i	$-0.362390\pi$
$439$	17.5569	0.837946	0.418973	+	0.907999i	$0.362390\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.9785	1.32930	0.664650	−	0.747155i	$-0.268581\pi$
$443$	27.9785	1.32930	0.664650	+	0.747155i	$0.268581\pi$
$444$	0	0
$445$	3.05863	0.144993
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.43459	0.445246	0.222623	−	0.974905i	$-0.428538\pi$
$449$	9.43459	0.445246	0.222623	+	0.974905i	$0.428538\pi$
$450$	0	0
$451$	3.39906	0.160056
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15.2457	−0.714730
$456$	0	0
$457$	12.1629	0.568957	0.284478	−	0.958682i	$-0.408180\pi$
$457$	12.1629	0.568957	0.284478	+	0.958682i	$0.408180\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14.7328	0.686176	0.343088	−	0.939303i	$-0.388527\pi$
$461$	14.7328	0.686176	0.343088	+	0.939303i	$0.388527\pi$
$462$	0	0
$463$	20.2423	0.940738	0.470369	−	0.882470i	$-0.344121\pi$
$463$	20.2423	0.940738	0.470369	+	0.882470i	$0.344121\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−34.5484	−1.59871	−0.799355	−	0.600859i	$-0.794825\pi$
$467$	−34.5484	−1.59871	−0.799355	+	0.600859i	$0.794825\pi$
$468$	0	0
$469$	36.8888	1.70337
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−33.2603	−1.52931
$474$	0	0
$475$	2.24914	0.103198
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.0214836	−0.000981610 0	−0.000490805	−	1.00000i	$-0.500156\pi$
$479$	−0.0214836	−0.000981610 0	−0.000490805		1.00000i	$0.500156\pi$
$480$	0	0
$481$	−35.9785	−1.64048
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.43965	−0.247002
$486$	0	0
$487$	−14.8387	−0.672406	−0.336203	−	0.941790i	$-0.609143\pi$
$487$	−14.8387	−0.672406	−0.336203	+	0.941790i	$0.609143\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.81904	0.217480	0.108740	−	0.994070i	$-0.465318\pi$
$491$	4.81904	0.217480	0.108740	+	0.994070i	$0.465318\pi$
$492$	0	0
$493$	19.2215	0.865695
$494$	0	0
$495$	0	0
$496$	0	0
$497$	76.5708	3.43467
$498$	0	0
$499$	40.0647	1.79354	0.896772	−	0.442493i	$-0.145906\pi$
$499$	40.0647	1.79354	0.896772	+	0.442493i	$0.145906\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.1449	1.47786	0.738928	−	0.673784i	$-0.235332\pi$
$503$	33.1449	1.47786	0.738928	+	0.673784i	$0.235332\pi$
$504$	0	0
$505$	8.52932	0.379550
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.9053	−1.54715	−0.773575	−	0.633705i	$-0.781533\pi$
$509$	−34.9053	−1.54715	−0.773575	+	0.633705i	$0.781533\pi$
$510$	0	0
$511$	37.8613	1.67488
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.32238	−0.410793
$516$	0	0
$517$	9.08785	0.399683
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.0180	0.657953	0.328976	−	0.944338i	$-0.393296\pi$
$521$	15.0180	0.657953	0.328976	+	0.944338i	$0.393296\pi$
$522$	0	0
$523$	−3.00344	−0.131331	−0.0656656	−	0.997842i	$-0.520917\pi$
$523$	−3.00344	−0.131331	−0.0656656	+	0.997842i	$0.520917\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.9294	−1.08594
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.27856	−0.142010
$534$	0	0
$535$	13.1449	0.568302
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−52.3741	−2.25591
$540$	0	0
$541$	5.90871	0.254035	0.127018	−	0.991900i	$-0.459459\pi$
$541$	5.90871	0.254035	0.127018	+	0.991900i	$0.459459\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.0406	0.430092
$546$	0	0
$547$	39.1138	1.67239	0.836193	−	0.548435i	$-0.184776\pi$
$547$	39.1138	1.67239	0.836193	+	0.548435i	$0.184776\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.9233	0.763559
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.59139	0.194543	0.0972717	−	0.995258i	$-0.468988\pi$
$557$	4.59139	0.194543	0.0972717	+	0.995258i	$0.468988\pi$
$558$	0	0
$559$	32.0812	1.35689
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.3208	0.519258	0.259629	−	0.965708i	$-0.416400\pi$
$563$	12.3208	0.519258	0.259629	+	0.965708i	$0.416400\pi$
$564$	0	0
$565$	3.70522	0.155880
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.1430	−1.05405	−0.527026	−	0.849849i	$-0.676693\pi$
$569$	−25.1430	−1.05405	−0.527026	+	0.849849i	$0.676693\pi$
$570$	0	0
$571$	−17.2197	−0.720623	−0.360311	−	0.932832i	$-0.617330\pi$
$571$	−17.2197	−0.720623	−0.360311	+	0.932832i	$0.617330\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	32.7880	1.36498	0.682491	−	0.730894i	$-0.260896\pi$
$577$	32.7880	1.36498	0.682491	+	0.730894i	$0.260896\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−60.1346	−2.49480
$582$	0	0
$583$	−18.4837	−0.765516
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.9018	1.23418	0.617090	−	0.786892i	$-0.288311\pi$
$587$	29.9018	1.23418	0.617090	+	0.786892i	$0.288311\pi$
$588$	0	0
$589$	−23.2457	−0.957822
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−0.656135	−0.0269442	−0.0134721	−	0.999909i	$-0.504288\pi$
$593$	−0.656135	−0.0269442	−0.0134721	+	0.999909i	$0.504288\pi$
$594$	0	0
$595$	−11.5259	−0.472515
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−41.2603	−1.68585	−0.842925	−	0.538031i	$-0.819168\pi$
$599$	−41.2603	−1.68585	−0.842925	+	0.538031i	$0.819168\pi$
$600$	0	0
$601$	−12.1008	−0.493604	−0.246802	−	0.969066i	$-0.579380\pi$
$601$	−12.1008	−0.493604	−0.246802	+	0.969066i	$0.579380\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.0586332	−0.00238378
$606$	0	0
$607$	−37.1475	−1.50777	−0.753886	−	0.657005i	$-0.771823\pi$
$607$	−37.1475	−1.50777	−0.753886	+	0.657005i	$0.771823\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.76567	−0.354621
$612$	0	0
$613$	−31.3415	−1.26587	−0.632935	−	0.774205i	$-0.718150\pi$
$613$	−31.3415	−1.26587	−0.632935	+	0.774205i	$0.718150\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.26213	−0.0508115	−0.0254057	−	0.999677i	$-0.508088\pi$
$617$	−1.26213	−0.0508115	−0.0254057	+	0.999677i	$0.508088\pi$
$618$	0	0
$619$	−31.1138	−1.25057	−0.625285	−	0.780396i	$-0.715017\pi$
$619$	−31.1138	−1.25057	−0.625285	+	0.780396i	$0.715017\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−14.6155	−0.585560
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−27.2001	−1.08454
$630$	0	0
$631$	−43.2648	−1.72234	−0.861172	−	0.508313i	$-0.830269\pi$
$631$	−43.2648	−1.72234	−0.861172	+	0.508313i	$0.830269\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.68879	0.0670175
$636$	0	0
$637$	50.5174	2.00157
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.01461	0.316558	0.158279	−	0.987394i	$-0.449405\pi$
$641$	8.01461	0.316558	0.158279	+	0.987394i	$0.449405\pi$
$642$	0	0
$643$	27.0061	1.06502	0.532509	−	0.846425i	$-0.321249\pi$
$643$	27.0061	1.06502	0.532509	+	0.846425i	$0.321249\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−14.6854	−0.577341	−0.288670	−	0.957429i	$-0.593213\pi$
$647$	−14.6854	−0.577341	−0.288670	+	0.957429i	$0.593213\pi$
$648$	0	0
$649$	−6.51289	−0.255653
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.4871	−0.449525	−0.224763	−	0.974414i	$-0.572161\pi$
$653$	−11.4871	−0.449525	−0.224763	+	0.974414i	$0.572161\pi$
$654$	0	0
$655$	8.99656	0.351525
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.9199	0.737014	0.368507	−	0.929625i	$-0.379869\pi$
$659$	18.9199	0.737014	0.368507	+	0.929625i	$0.379869\pi$
$660$	0	0
$661$	−47.3707	−1.84251	−0.921253	−	0.388963i	$-0.872833\pi$
$661$	−47.3707	−1.84251	−0.921253	+	0.388963i	$0.872833\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10.7474	−0.416767
$666$	0	0
$667$	−7.96896	−0.308560
$668$	0	0
$669$	0	0
$670$	0	0
$671$	27.2863	1.05338
$672$	0	0
$673$	38.2130	1.47300	0.736502	−	0.676435i	$-0.236476\pi$
$673$	38.2130	1.47300	0.736502	+	0.676435i	$0.236476\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−47.8156	−1.83770	−0.918852	−	0.394603i	$-0.870882\pi$
$677$	−47.8156	−1.83770	−0.918852	+	0.394603i	$0.870882\pi$
$678$	0	0
$679$	25.9931	0.997525
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.2491	1.00440	0.502198	−	0.864753i	$-0.332525\pi$
$683$	26.2491	1.00440	0.502198	+	0.864753i	$0.332525\pi$
$684$	0	0
$685$	19.5569	0.747231
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.8284	0.679208
$690$	0	0
$691$	18.0552	0.686852	0.343426	−	0.939180i	$-0.388413\pi$
$691$	18.0552	0.686852	0.343426	+	0.939180i	$0.388413\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.3319	−0.581573
$696$	0	0
$697$	−2.47862	−0.0938843
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.0766789	−0.00289612	−0.00144806	−	0.999999i	$-0.500461\pi$
$701$	−0.0766789	−0.00289612	−0.00144806	+	0.999999i	$0.500461\pi$
$702$	0	0
$703$	−25.3630	−0.956582
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−40.7570	−1.53282
$708$	0	0
$709$	−5.48024	−0.205815	−0.102907	−	0.994691i	$-0.532814\pi$
$709$	−5.48024	−0.205815	−0.102907	+	0.994691i	$0.532814\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.3354	0.387063
$714$	0	0
$715$	−10.5535	−0.394678
$716$	0	0
$717$	0	0
$718$	0	0
$719$	17.5811	0.655663	0.327832	−	0.944736i	$-0.393682\pi$
$719$	17.5811	0.655663	0.327832	+	0.944736i	$0.393682\pi$
$720$	0	0
$721$	44.5466	1.65900
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.96896	0.295960
$726$	0	0
$727$	−28.2993	−1.04956	−0.524781	−	0.851237i	$-0.675853\pi$
$727$	−28.2993	−1.04956	−0.524781	+	0.851237i	$0.675853\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24.2536	0.897053
$732$	0	0
$733$	13.9836	0.516495	0.258248	−	0.966079i	$-0.416855\pi$
$733$	13.9836	0.516495	0.258248	+	0.966079i	$0.416855\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.5354	0.940610
$738$	0	0
$739$	−12.9509	−0.476407	−0.238204	−	0.971215i	$-0.576559\pi$
$739$	−12.9509	−0.476407	−0.238204	+	0.971215i	$0.576559\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.64820	−0.0604666	−0.0302333	−	0.999543i	$-0.509625\pi$
$743$	−1.64820	−0.0604666	−0.0302333	+	0.999543i	$0.509625\pi$
$744$	0	0
$745$	−9.80605	−0.359266
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−62.8122	−2.29511
$750$	0	0
$751$	−34.6233	−1.26342	−0.631711	−	0.775204i	$-0.717647\pi$
$751$	−34.6233	−1.26342	−0.631711	+	0.775204i	$0.717647\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.0552	0.511521
$756$	0	0
$757$	15.9183	0.578559	0.289280	−	0.957245i	$-0.406584\pi$
$757$	15.9183	0.578559	0.289280	+	0.957245i	$0.406584\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.58795	−0.0575631	−0.0287816	−	0.999586i	$-0.509163\pi$
$761$	−1.58795	−0.0575631	−0.0287816	+	0.999586i	$0.509163\pi$
$762$	0	0
$763$	−47.9785	−1.73694
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.28200	0.226830
$768$	0	0
$769$	50.6302	1.82577	0.912885	−	0.408217i	$-0.133849\pi$
$769$	50.6302	1.82577	0.912885	+	0.408217i	$0.133849\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.37758	−0.193418	−0.0967090	−	0.995313i	$-0.530832\pi$
$773$	−5.37758	−0.193418	−0.0967090	+	0.995313i	$0.530832\pi$
$774$	0	0
$775$	−10.3354	−0.371257
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.31121	−0.0828077
$780$	0	0
$781$	53.0043	1.89664
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.28018	−0.0813830
$786$	0	0
$787$	6.42666	0.229086	0.114543	−	0.993418i	$-0.463460\pi$
$787$	6.42666	0.229086	0.114543	+	0.993418i	$0.463460\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17.7052	−0.629525
$792$	0	0
$793$	−26.3189	−0.934613
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.2295	0.504034	0.252017	−	0.967723i	$-0.418906\pi$
$797$	14.2295	0.504034	0.252017	+	0.967723i	$0.418906\pi$
$798$	0	0
$799$	−6.62692	−0.234444
$800$	0	0
$801$	0	0
$802$	0	0
$803$	26.2086	0.924880
$804$	0	0
$805$	4.77846	0.168418
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.5551	1.60163	0.800816	−	0.598911i	$-0.204400\pi$
$809$	45.5551	1.60163	0.800816	+	0.598911i	$0.204400\pi$
$810$	0	0
$811$	17.1043	0.600612	0.300306	−	0.953843i	$-0.402911\pi$
$811$	17.1043	0.600612	0.300306	+	0.953843i	$0.402911\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.5569	0.474878
$816$	0	0
$817$	22.6155	0.791218
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.9018	1.67179	0.835893	−	0.548893i	$-0.184950\pi$
$821$	47.9018	1.67179	0.835893	+	0.548893i	$0.184950\pi$
$822$	0	0
$823$	19.5309	0.680806	0.340403	−	0.940280i	$-0.389437\pi$
$823$	19.5309	0.680806	0.340403	+	0.940280i	$0.389437\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19.5879	0.681140	0.340570	−	0.940219i	$-0.389380\pi$
$827$	19.5879	0.681140	0.340570	+	0.940219i	$0.389380\pi$
$828$	0	0
$829$	−30.1560	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$829$	−30.1560	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	38.1916	1.32326
$834$	0	0
$835$	12.8647	0.445201
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−21.3224	−0.736130	−0.368065	−	0.929800i	$-0.619980\pi$
$839$	−21.3224	−0.736130	−0.368065	+	0.929800i	$0.619980\pi$
$840$	0	0
$841$	34.5044	1.18981
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.82066	−0.0970337
$846$	0	0
$847$	0.280176	0.00962696
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.2767	0.386562
$852$	0	0
$853$	46.3741	1.58782	0.793910	−	0.608035i	$-0.208042\pi$
$853$	46.3741	1.58782	0.793910	+	0.608035i	$0.208042\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	50.9966	1.74201	0.871005	−	0.491275i	$-0.163469\pi$
$857$	50.9966	1.74201	0.871005	+	0.491275i	$0.163469\pi$
$858$	0	0
$859$	12.6872	0.432881	0.216440	−	0.976296i	$-0.430555\pi$
$859$	12.6872	0.432881	0.216440	+	0.976296i	$0.430555\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.5827	1.58569	0.792847	−	0.609421i	$-0.208598\pi$
$863$	46.5827	1.58569	0.792847	+	0.609421i	$0.208598\pi$
$864$	0	0
$865$	11.3224	0.384973
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.6302	−0.834561
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.77846	−0.161541
$876$	0	0
$877$	0.270624	0.00913833	0.00456916	−	0.999990i	$-0.498546\pi$
$877$	0.270624	0.00913833	0.00456916	+	0.999990i	$0.498546\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.5389	0.759354	0.379677	−	0.925119i	$-0.376035\pi$
$881$	22.5389	0.759354	0.379677	+	0.925119i	$0.376035\pi$
$882$	0	0
$883$	−48.3043	−1.62557	−0.812785	−	0.582564i	$-0.802050\pi$
$883$	−48.3043	−1.62557	−0.812785	+	0.582564i	$0.802050\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.9379	0.400836	0.200418	−	0.979710i	$-0.435770\pi$
$887$	11.9379	0.400836	0.200418	+	0.979710i	$0.435770\pi$
$888$	0	0
$889$	−8.06980	−0.270653
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6.17934	−0.206784
$894$	0	0
$895$	20.9345	0.699763
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−82.3622	−2.74693
$900$	0	0
$901$	13.4784	0.449031
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.93793	0.0644189
$906$	0	0
$907$	−28.8268	−0.957177	−0.478589	−	0.878039i	$-0.658851\pi$
$907$	−28.8268	−0.957177	−0.478589	+	0.878039i	$0.658851\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−54.5726	−1.80807	−0.904035	−	0.427458i	$-0.859409\pi$
$911$	−54.5726	−1.80807	−0.904035	+	0.427458i	$0.859409\pi$
$912$	0	0
$913$	−41.6267	−1.37764
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−42.9897	−1.41964
$918$	0	0
$919$	30.2277	0.997118	0.498559	−	0.866856i	$-0.333863\pi$
$919$	30.2277	0.997118	0.498559	+	0.866856i	$0.333863\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−51.1252	−1.68281
$924$	0	0
$925$	−11.2767	−0.370777
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−40.1706	−1.31796	−0.658978	−	0.752162i	$-0.729011\pi$
$929$	−40.1706	−1.31796	−0.658978	+	0.752162i	$0.729011\pi$
$930$	0	0
$931$	35.6121	1.16714
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.97852	−0.260925
$936$	0	0
$937$	−10.3258	−0.337330	−0.168665	−	0.985673i	$-0.553946\pi$
$937$	−10.3258	−0.337330	−0.168665	+	0.985673i	$0.553946\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	43.9096	1.43141	0.715706	−	0.698402i	$-0.246105\pi$
$941$	43.9096	1.43141	0.715706	+	0.698402i	$0.246105\pi$
$942$	0	0
$943$	1.02760	0.0334632
$944$	0	0
$945$	0	0
$946$	0	0
$947$	58.4914	1.90072	0.950358	−	0.311160i	$-0.100717\pi$
$947$	58.4914	1.90072	0.950358	+	0.311160i	$0.100717\pi$
$948$	0	0
$949$	−25.2794	−0.820605
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.5309	−0.697455	−0.348728	−	0.937224i	$-0.613386\pi$
$953$	−21.5309	−0.697455	−0.348728	+	0.937224i	$0.613386\pi$
$954$	0	0
$955$	11.7440	0.380026
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−93.4519	−3.01772
$960$	0	0
$961$	75.8199	2.44580
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.29317	0.106011
$966$	0	0
$967$	−14.9629	−0.481173	−0.240586	−	0.970628i	$-0.577340\pi$
$967$	−14.9629	−0.481173	−0.240586	+	0.970628i	$0.577340\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.0777	1.31825	0.659124	−	0.752035i	$-0.270927\pi$
$971$	41.0777	1.31825	0.659124	+	0.752035i	$0.270927\pi$
$972$	0	0
$973$	73.2630	2.34870
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.76041	−0.184292	−0.0921459	−	0.995746i	$-0.529373\pi$
$977$	−5.76041	−0.184292	−0.0921459	+	0.995746i	$0.529373\pi$
$978$	0	0
$979$	−10.1173	−0.323349
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.1223	0.450432	0.225216	−	0.974309i	$-0.427691\pi$
$983$	14.1223	0.450432	0.225216	+	0.974309i	$0.427691\pi$
$984$	0	0
$985$	6.73281	0.214525
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−10.0552	−0.319737
$990$	0	0
$991$	30.4458	0.967142	0.483571	−	0.875305i	$-0.339340\pi$
$991$	30.4458	0.967142	0.483571	+	0.875305i	$0.339340\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.82066	0.311336
$996$	0	0
$997$	1.17590	0.0372411	0.0186206	−	0.999827i	$-0.494073\pi$
$997$	1.17590	0.0372411	0.0186206	+	0.999827i	$0.494073\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bm.1.1		3
3.2	odd	2		2760.2.a.r.1.1	✓	3
12.11	even	2		5520.2.a.bx.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.r.1.1	✓	3	3.2	odd	2
5520.2.a.bx.1.3		3	12.11	even	2
8280.2.a.bm.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.77846 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.77846 q^{7} -3.30777 q^{11} +3.19051 q^{13} +2.41205 q^{17} +2.24914 q^{19} -1.00000 q^{23} +1.00000 q^{25} +7.96896 q^{29} -10.3354 q^{31} -4.77846 q^{35} -11.2767 q^{37} -1.02760 q^{41} +10.0552 q^{43} -2.74742 q^{47} +15.8337 q^{49} +5.58795 q^{53} -3.30777 q^{55} +1.96896 q^{59} -8.24914 q^{61} +3.19051 q^{65} -7.71982 q^{67} -16.0242 q^{71} -7.92332 q^{73} +15.8061 q^{77} +12.5845 q^{83} +2.41205 q^{85} +3.05863 q^{89} -15.2457 q^{91} +2.24914 q^{95} -5.43965 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 6 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 6 * q^7	\( 3 q + 3 q^{5} - 6 q^{7} - 2 q^{11} + 6 q^{17} - 2 q^{19} - 3 q^{23} + 3 q^{25} + 6 q^{29} - 6 q^{31} - 6 q^{35} - 8 q^{37} + 14 q^{41} - 4 q^{43} + 18 q^{47} + 5 q^{49} + 18 q^{53} - 2 q^{55} - 12 q^{59} - 16 q^{61} - 14 q^{67} + 4 q^{71} + 22 q^{77} + 4 q^{83} + 6 q^{85} + 10 q^{89} - 2 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 6 * q^7 - 2 * q^11 + 6 * q^17 - 2 * q^19 - 3 * q^23 + 3 * q^25 + 6 * q^29 - 6 * q^31 - 6 * q^35 - 8 * q^37 + 14 * q^41 - 4 * q^43 + 18 * q^47 + 5 * q^49 + 18 * q^53 - 2 * q^55 - 12 * q^59 - 16 * q^61 - 14 * q^67 + 4 * q^71 + 22 * q^77 + 4 * q^83 + 6 * q^85 + 10 * q^89 - 2 * q^91 - 2 * q^95 + 2 * q^97

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